Transition asymptotics of Soliton equations

The beautiful, intriguing nature of soliton equations (or completely integrable, nonlinear, partial differential equations) lies in the fact that one can write down the explicit solution using the inverse scattering transform. This is in sharp contrast to other ordinary or partial differential equations, where only the simplest equations are solvable by explicit formulae and the vast majority is left to numerical approximation schemes using computers. Think of water waves and the original discovery of solitary waves or solitons: in 1834 John Scott Russell observed the bow wave of a boat which had just stopped. The wave continued forward for a mile without change of form or diminution of speed. Nonlinear and dispersive effects had balanced each other and produced a wave which propagated without change of its shape. This phenomenon was explained theoretically by the Korteweg-de Vries (KdV) equation in 1895. Not until 1967, when interest in KdV was renewed in the wake of the famous FPU computer experiment, a mathematical solution for the KdV equation was found by Gardner, Greene, Kruskal, and Miura, using inverse scattering theory from quantum mechanics. Shortly after, Peter Lax introduced a unified approach, which extended this method to other soliton equations. Since then, this fascinating area has drawn enormous interest. The aim of the project is to study the stability of certain soliton equations under small perturbations of the initial data and to analyze the explicit solution for large times. We will first consider the Toda lattice, which is the prototypical example of a discrete-space, continuous-time soliton equation. We will examine asymptotic regimes in space-time and calculate the local waveforms. Not even in the simplest case of the Toda lattice with decaying background is the asymptotic behavior of the solution completely understood. The solution eventually decomposes into several components: a number of solitons (corresponding to the eigenvalues of the underlying Lax operator), a dispersive radiation part (corresponding to the continuous spectrum of the Lax operator), a Painlevé regime, and two unknown transition regimes which bridge the dispersive and Painlevé regimes. Our goal is to give a rigorous mathematical description for the transition and collisionless shock regions. Moreover, we want to understand and explain what happens for step-like initial data when the perturbation is different on the two sides. The Toda lattice will be used as a model to develop ideas and methods applicable to all soliton equations in one space dimension. For example, the KdV equation emerges in a certain scaling from the Toda lattice. The Toda lattice itself is a simple model in solid state physics and used for modeling Langmuir oscillations in plasma physics, conducting polymers, soliton communication channels, et cetera and we expect implications of our results there.

In this project, we studied the long-time asymptotic behavior of solutions of soliton equations. We were interested in the stability of such equations under small perturbations or disturbances of the initial data. The models we considered are the Toda lattice equation (the prototypical example of a discrete-space, continuous-time soliton equation) and the celebrated Korteweg-de Vries equation, which models waves on shallow water surfaces. We subjected the equations to shock-type initial conditions and investigated what happens to the solution for large times. The intriguing nature of soliton equations lies in the fact that one can write down the explicit solution using the inverse scattering transform. By applying the nonlinear steepest descent analysis, with modifications and extensions convenient for our setting, we were able to analyze this explicit solution for large times and give a complete answer in the case of the Toda shock problem. The most interesting results are in the regime when the space variable n and the time variable t go to infinity with the ratio n / t close to a constant. We had previously established in FWF V120 that there are five principal regions in the (n, t) half-plane with essentially different qualitative behavior of the solution of the Toda shock problem: the (left and right) soliton regions, where the solution is asymptotically close to the constant (left or right) background solution and a number of solitons, that is, pulse-like waves which spread in time without changing its size and shape; the (left and right) modulation regions, where the solution undergoes a modulation and is asymptotically close to a modulated single-phase quasi-periodic Toda solution; and the elliptic region or middle region, where the solution is asymptotically close to a quasi-periodic Toda solution. In this project we made the analysis rigorous and computed the error terms, that is, the "closeness" of the solution and the asymptotic expansion. We allowed more general initial data in shock position and investigated the influence of eigenvalues in the middle region and resonances on the solution. For the Korteweg-de Vries shock problem, the solution eventually splits into a decaying dispersive tail, a dispersive shock wave, and a number of solitons. The principal regions and corresponding asymptotics are well understood, but the regions do not overlap. It is a delicate task to improve the domain of validity of these formulas to achieve overlap. We refined the Riemann-Hilbert analysis in the soliton region to both increase the domain of validity and weaken the decay and smoothness requirements for the initial data. For possible applications, the most interesting result is that we justify the soliton asymptotics of step-like solutions in a larger region than previously known and for essentially larger classes of initial data.